Erdős–Burgess constant in commutative rings

Abstract

Let $$\mathcal {S}$$ be a nonempty semigroup endowed with a binary associative operation $$*$$ . An element e of $$\mathcal {S}$$ is said to be idempotent if $$e*e=e$$ . Originated by one question of P. Erdős to D.A. Burgess: If $$\mathcal {S}$$ is a finite semigroup of order n, does any $$\mathcal {S}$$ -valued sequence T of length n contain a nonempty subsequence the product of whose terms, in some order, is idempotent?, we make a study of the associated invariant, denoted $$\mathrm{I}(\mathcal {S})$$ and called Erdős–Burgess constant which is the smallest positive integer $$\ell $$ such that any $$\mathcal {S}$$ -valued sequence T of length $$\ell $$ contains a nonempty subsequence the product of whose terms, in some order, is an idempotent. Let $$\mathcal {S}_R$$ be the multiplicative semigroup of any commutative unitary ring R. We prove that $$\mathrm{I}(\mathcal {S}_R)$$ is finite if and only if R is finite, provided that the quotient ring R/J(R) of R modulo its Jacobson radical J(R) is not a direct product of an infinite Boolean unitary ring and finitely many finite fields. As a consequence, if R is Noetherian, then $$\mathrm{I}(\mathcal {S}_R)$$ is finite if and only if R is finite.